Here and throughout the paper, \(\dim \)  refers to Hausdorff dimension. Very recently, Orponen and Shmerkin [1]} obtained a quantitatively small (but qualitatively very significant) improvement to the RHS of (REF ) whenever \(0<s<\dim A\)  and \(s<1\) . In a slightly different direction, Bourgain [2]} obtained a large improvement over (REF ) when \(s\)  is close to \((\dim A)/2\) . As a special case of his result, we have the following.
